Numerical methods for solving a two-dimensional variable-order anomalous subdiffusion equation

Chen, Chang-Ming; Liu, F.; Anh, V.; Turner, I.

doi:10.1090/S0025-5718-2011-02447-6

Numerical methods for solving a two-dimensional variable-order anomalous subdiffusion equation
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by Chang-Ming Chen, F. Liu, V. Anh and I. Turner PDF

Math. Comp. 81 (2012), 345-366 Request permission

Abstract:

Anomalous subdiffusion equations have in recent years received much attention. In this paper, we consider a two-dimensional variable-order anomalous subdiffusion equation. Two numerical methods (the implicit and explicit methods) are developed to solve the equation. Their stability, convergence and solvability are investigated by the Fourier method. Moreover, the effectiveness of our theoretical analysis is demonstrated by some numerical examples.

References

Boris Baeumer, Mihály Kovács, and Mark M. Meerschaert, Numerical solutions for fractional reaction-diffusion equations, Comput. Math. Appl. 55 (2008), no. 10, 2212–2226. MR 2413687, DOI 10.1016/j.camwa.2007.11.012
Chang-Ming Chen, F. Liu, I. Turner, and V. Anh, A Fourier method for the fractional diffusion equation describing sub-diffusion, J. Comput. Phys. 227 (2007), no. 2, 886–897. MR 2442379, DOI 10.1016/j.jcp.2007.05.012
Chang-ming Chen, F. Liu, and K. Burrage, Finite difference methods and a Fourier analysis for the fractional reaction-subdiffusion equation, Appl. Math. Comput. 198 (2008), no. 2, 754–769. MR 2405817, DOI 10.1016/j.amc.2007.09.020
Chang-Ming Chen, F. Liu, and V. Anh, Numerical analysis of the Rayleigh-Stokes problem for a heated generalized second grade fluid with fractional derivatives, Appl. Math. Comput. 204 (2008), no. 1, 340–351. MR 2458372, DOI 10.1016/j.amc.2008.06.052
Chang-Ming Chen, F. Liu, and V. Anh, A Fourier method and an extrapolation technique for Stokes’ first problem for a heated generalized second grade fluid with fractional derivative, J. Comput. Appl. Math. 223 (2009), no. 2, 777–789. MR 2478879, DOI 10.1016/j.cam.2008.03.001
Chang-Ming Chen and F. Liu, A numerical approximation method for solving a three-dimensional space Galilei invariant fractional advection-diffusion equation, J. Appl. Math. Comput. 30 (2009), no. 1-2, 219–236. MR 2496613, DOI 10.1007/s12190-008-0168-7
Chang-Ming Chen, Fawang Liu, Ian Turner, and Vo Anh, Numerical schemes and multivariate extrapolation of a two-dimensional anomalous sub-diffusion equation, Numer. Algorithms 54 (2010), no. 1, 1–21. MR 2610319, DOI 10.1007/s11075-009-9320-1
Carlos F. M. Coimbra, Mechanics with variable-order differential operators, Ann. Phys. 12 (2003), no. 11-12, 692–703. MR 2020716, DOI 10.1002/andp.200310032
Kristian P. Evans and Niels Jacob, Feller semigroups obtained by variable order subordination, Rev. Mat. Complut. 20 (2007), no. 2, 293–307. MR 2351111, DOI 10.5209/rev_{R}EMA.2007.v20.n2.16482
Niels Jacob and Hans-Gerd Leopold, Pseudo-differential operators with variable order of differentiation generating Feller semigroups, Integral Equations Operator Theory 17 (1993), no. 4, 544–553. MR 1243995, DOI 10.1007/BF01200393
Koji Kikuchi and Akira Negoro, On Markov process generated by pseudodifferential operator of variable order, Osaka J. Math. 34 (1997), no. 2, 319–335. MR 1483853
T. A. M. Langlands and B. I. Henry, The accuracy and stability of an implicit solution method for the fractional diffusion equation, J. Comput. Phys. 205 (2005), no. 2, 719–736. MR 2135000, DOI 10.1016/j.jcp.2004.11.025
T. A. M. Langlands, B. I. Henry, and S. L. Wearne, Anomalous subdiffusion with multispecies linear reaction dynamics, Phys. Rev. E (3) 77 (2008), no. 2, 021111, 9. MR 2453275, DOI 10.1103/PhysRevE.77.021111
Hans-Gerd Leopold, Embedding of function spaces of variable order of differentiation in function spaces of variable order of integration, Czechoslovak Math. J. 49(124) (1999), no. 3, 633–644. MR 1708338, DOI 10.1023/A:1022483721944
R. Lin, F. Liu, V. Anh, and I. Turner, Stability and convergence of a new explicit finite-difference approximation for the variable-order nonlinear fractional diffusion equation, Appl. Math. Comput. 212 (2009), no. 2, 435–445. MR 2531970, DOI 10.1016/j.amc.2009.02.047
Yumin Lin and Chuanju Xu, Finite difference/spectral approximations for the time-fractional diffusion equation, J. Comput. Phys. 225 (2007), no. 2, 1533–1552. MR 2349193, DOI 10.1016/j.jcp.2007.02.001
F. Liu, C. Yang, and K. Burrage, Numerical method and analytic technique of the modified anomalous subdiffusion equation with a nonlinear source term, J. Comput. Appl. Math. 231 (2009), no. 1, 160–176. MR 2532659, DOI 10.1016/j.cam.2009.02.013
F. Liu, P. Zhuang, V. Anh, I. Turner, and K. Burrage, Stability and convergence of the difference methods for the space-time fractional advection-diffusion equation, Appl. Math. Comput. 191 (2007), no. 1, 12–20. MR 2385498, DOI 10.1016/j.amc.2006.08.162
Q. Liu, F. Liu, I. Turner, and V. Anh, Approximation of the Lévy-Feller advection-dispersion process by random walk and finite difference method, J. Comput. Phys. 222 (2007), no. 1, 57–70. MR 2298036, DOI 10.1016/j.jcp.2006.06.005
C. F. Lorenzo, T. T. Hartley, Initialization, conceptualization and application in the generalized fractional calculus, NASA/TP-1998-208-208415, 1998.
Carl F. Lorenzo and Tom T. Hartley, Variable order and distributed order fractional operators, Nonlinear Dynam. 29 (2002), no. 1-4, 57–98. Fractional order calculus and its applications. MR 1926468, DOI 10.1023/A:1016586905654
M. Meerschaert, A. Zoia, The Monte Carlo and fractional kinetics approaches to the underground anomalous subdiffusion of contaminants, Ann. Nucl. Energy, 33 (2006), pp. 223–235.
Mark M. Meerschaert, Hans-Peter Scheffler, and Charles Tadjeran, Finite difference methods for two-dimensional fractional dispersion equation, J. Comput. Phys. 211 (2006), no. 1, 249–261. MR 2168877, DOI 10.1016/j.jcp.2005.05.017
Charles Tadjeran and Mark M. Meerschaert, A second-order accurate numerical method for the two-dimensional fractional diffusion equation, J. Comput. Phys. 220 (2007), no. 2, 813–823. MR 2284325, DOI 10.1016/j.jcp.2006.05.030
Igor Podlubny, Fractional differential equations, Mathematics in Science and Engineering, vol. 198, Academic Press, Inc., San Diego, CA, 1999. An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. MR 1658022
Lynnette E. S. Ramirez and Carlos F. M. Coimbra, A variable order constitutive relation for viscoelasticity, Ann. Phys. 16 (2007), no. 7-8, 543–552. MR 2320863, DOI 10.1002/andp.200710246
T. V. Ratto, M. L. Longo, Anomalous subdiffusion in heterogeneous lipid bilayers, Langmuir, 19 (2003), pp, 1788-1793.
M. D. Ruiz-Medina, V. V. Anh, and J. M. Angulo, Fractional generalized random fields of variable order, Stochastic Anal. Appl. 22 (2004), no. 3, 775–799. MR 2047278, DOI 10.1081/SAP-120030456
Stefan G. Samko and Bertram Ross, Integration and differentiation to a variable fractional order, Integral Transform. Spec. Funct. 1 (1993), no. 4, 277–300. MR 1421643, DOI 10.1080/10652469308819027
M. J. Saxton, Anomalous subdiffusion in fluorescence photobleaching recovery: A Monte Carlo study, Biophys. J., 81 (2001), pp. 2226-2240.
P. Schwille, J. Korlach, W. W. Webb, Anomalous subdiffusion of proteins and lipids in membranes observed by fluorescence correlation spectroscopy, Biophys. J., 76 (1999), A391-A391.
C. M. Soon, C. F. M. Coimbra, M. H. Kobayashi, Variable viscoelasticity operator, Annalen der Physik, 14 (2005), pp. 378-389.
W. Tan, Chaoqi Fu, Ceji Fu, An anomalous subdiffusion model for calcium spark in cardiac myocytes, Appl. Phys. Lett. 91, 183901 (2007), doi:10.1063/1.2805208 .
M. Weiss, M. Elsner, F. Kartberg, T. Nilsson, Anomalous subdiffusion is a measure for cytoplasmic crowding in living cells, Biophys. J., 87 (2004), pp. 3518-3524.
Q. Yu, F. Liu, V. Anh, and I. Turner, Solving linear and non-linear space-time fractional reaction-diffusion equations by the Adomian decomposition method, Internat. J. Numer. Methods Engrg. 74 (2008), no. 1, 138–158. MR 2398497, DOI 10.1002/nme.2165
S. B. Yuste and L. Acedo, An explicit finite difference method and a new von Neumann-type stability analysis for fractional diffusion equations, SIAM J. Numer. Anal. 42 (2005), no. 5, 1862–1874. MR 2139227, DOI 10.1137/030602666
S. B. Yuste, Weighted average finite difference methods for fractional diffusion equations, J. Comput. Phys. 216 (2006), no. 1, 264–274. MR 2223444, DOI 10.1016/j.jcp.2005.12.006
P. Zhuang and F. Liu, Implicit difference approximation for the two-dimensional space-time fractional diffusion equation, J. Appl. Math. Comput. 25 (2007), no. 1-2, 269–282. MR 2345431, DOI 10.1007/BF02832352
P. Zhuang, F. Liu, V. Anh, and I. Turner, Numerical methods for the variable-order fractional advection-diffusion equation with a nonlinear source term, SIAM J. Numer. Anal. 47 (2009), no. 3, 1760–1781. MR 2505873, DOI 10.1137/080730597